1. Each principal component $w$ is a direction.
Its coordinate for a point is the dot product of that point with $w$.
The sign of this dot product tells us on which "side" of the origin the point lies when measured along $w$.
2. To visualize the sign, draw the line perpendicular to $w$ through the origin.
This line divides the plane into two half-planes:
- The half-plane that contains $w$ itself gives positive values of $w^T x$.
- The opposite half-plane gives negative values.
3. Apply this to each vector.
- For $w_1$ : check whether $P$ and $Q$ lie on the same side as the arrow of $w_1$ (positive) or the opposite side (negative).
- For $\mathrm{w}_2$ : do the same with the perpendicular line to $\mathrm{w}_2$.
Following this rule:
- P lies opposite $w_1 \rightarrow w_1$ coordinate negative; same side as $w_2 \rightarrow w_2$ coordinate positive.
- Q lies same side as $\mathrm{w}_1 \rightarrow \mathrm{w}_1$ coordinate positive; opposite $\mathrm{w}_2 \rightarrow \mathrm{w}_2$ coordinate negative.
That's why the signs come out as:
- $\mathrm{P}=(-,+)$
- $\mathrm{Q}=(+,-)$
